A new solution method for a class of large dimension rank-two nonconvex programs

Low rank problems are nonlinear minimization problems in which the objective function, by means of a suitable linear transformation of the variables, depends on very few variables. These problems often arise in quantitative management science applications, for example, in location models, transportation problems, production planning, data envelopment analysis, and multiobjective programs. They are usually approached by means of outer approximation, branch and bound, branch and select, and optimal level solution methods. The paper studies, from both a theoretical and an algorithmic point of view, a class of large dimension rank-two nonconvex problems having a polyhedral feasible region and $f(x)=phi(c^Tx+c_0,d^Tx+d_0)$ as the objective function. The proposed solution algorithm unifies a new partitioning method, an outer approximation approach, and a mixed method. The results of a computational test are provided to compare these three approaches with the optimal level solutions method. In particular, the new partitioning method performs very well in solving large problems.